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After weeks of rancor and relatively little direct engagement, Mr. Obama has expanded his recent aggressive attempts to work around GOP leaders and make his pitch for a deficit-reduction plan directly to rank-and-file Republican lawmakers.

The president followed his Wednesday dinner with 12 GOP senators by lunching Thursday with Rep. Paul Ryan (R., Wis.), the chairman of the House Budget Committee, and Rep. Chris Van Hollen (D., Md.), the senior Democrat on the panel.

Both Republicans and administration officials described the discussions as productive and even pleasant. But while the lawmakers' mood may be improving, fundamental disagreements over taxes and spending persist, and lawmakers in both parties acknowledged that any significant deal would be months away at best. "I don't think anyone left there with any anticipation that over the next month or six weeks anything is going to happen," said Sen. Bob Corker (R., Tenn.), who attended the dinner. "I think the goal would be to make something happen over the next four or five months."

With across-the-board budget cuts known as the sequester having taken effect, and both the House and Senate writing legislation to avoid a partial government shutdown later this month, many lawmakers have pointed to this summer—when the government must again raise its borrowing limit—as a possible target for reaching a big deal.

House Speaker John Boehner noted Mr. Obama's outreach effort and said he hoped "something will come out of it." But he said Thursday he wouldn't support raising the debt ceiling without matching spending cuts.

Both the president and some GOP members have indicated a willingness to discuss a deficit-reduction approach that would involve overhauling the tax code and slowing the growth of spending on entitlement programs such as Medicare and Social Security. But, as Sen. Lindsey Graham (R., S.C.) noted, "The details may trip us up."

Among the potential obstacles: The White House wants tax changes that result in a net rise in revenue, which many GOP lawmakers oppose. Many Republicans want bigger changes to entitlement programs than Democrats favor. And an effort to splinter the GOP could backfire, particularly if liberal Democrats revolt at any proposal they see as too Republican-friendly.

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On the day President Obama invited a group of Republican Senators to a private dinner, Sen. John McCain tells WSJ's Jerry Seib that such overtures are being "well received" and are a sign that the president "now is interested in outreach and dialogue."

Both sides offered cautionary notes Thursday. "It's going to be hard," White House Press Secretary Jay Carney said. "I do not want to in any way to suggest that this is a sure thing. It is far from that."

Mr. Obama's current approach marks a shift from his recent public criticism of GOP lawmakers as he tried, unsuccessfully, to pressure them to negotiate over the sequester. His overtures to the Republican rank-and-file represents a departure from his failed efforts in 2011 and 2012 to broker large-scale deficit-reduction deals by negotiating primarily with Mr. Boehner.

Now, Mr. Obama is trying to bypass the congressional leadership in hopes of brokering a deal with enough Senate Republicans so that body can pass a bipartisan bill and send it to the House.

Mr. Obama has focused his attention largely on Republicans this week, a move that House Minority Leader Nancy Pelosi (D., Calif.) described as useful to understand what measures Republican members might support. "I think it's important that they all get to know each other better," she said.Some Republicans said the president has some catching up to do, after making little effort to build relationships during his first term—a claim the White House disputes. At the same time, some said they are willing to look forward, not back.

"As an avid golfer, the president knows that one of the most important parts of the swing is the follow-through," Sen. Mike Johanns (R., Neb.) said in a statement after dining with Mr. Obama. "The same holds true here as well."

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